THE STANDARDIZATION OF WEIGHTS BY RICHARDS' METHOD WALDOL. SEMON, UNIVERSITY OP WASHINGTON, SEATTLE, WASHINGTON
Richards' method for the standardization of weights' has been found in this laboratory t o excel1 both in simplicity and accuracy. There are, however, certain difficulties which arise in the teaching of this method. (1) The data form should include all of the experimental data and calculations, tabulated upon a single sheet. (2) There should be ample opportunity afforded for checking the standardization. (3) There should be somewhere available a mathematical proof of the accuracy of the method.
-
Weights placed on the left-handpan
Readings of rider an beam With two or more weights taken together
weight in
With
terms of the
single weight
Standard
preliminary
Proportional Correcparts of tions in the final Mg standard
-
If weighings are done by the method of equal swings as recommended by Wells2it becomes extremely easy t o devise a data sheet having the proper requisites. Experience has shown that with such a sheet, very few errors are made ' Richards. 3. Am. Chm. Soc., 22 144 (1900). a I.Am. Ckm. Sac., 42 411 (1920).
VOL.2, NO. 2 STAND~RDIZATION OW WEIGHTSBY RICHARDS' METHOD
133
in the first three columns. Numerical results in the derived columns can be checked over and corrected if necessary, avoiding in most cases, the entire restandardization of the set. Such a procedure of rechecking is impossible if columns two and three have been omitted and the correction determined by inspection as in the original article by Richards.* Furthermore, this scheme makes it possible to check any one weigbt separately without going through the entire set. An important part of any calibration is the interchecking to make sure that the figures are consistent among themselves and that no cumulative errors have crept in. Hopkins, Zinn and RogersSaccomplish this by comparison of the derived values a t certain intervals with Bureau of Standards weights. By altering slightly, this procedure may be used to furnish an indication of the excellence of the calibration of any set. If, for instance, a Bureau of Standards' 50 g. weight is given to the student as a final standard to which all of the weights are to be referred, then all of the weights are corrected in terms of the international kilogram. The check may take the form of an "unknown" exercise. A weight of approximately 49.99 g., the mass of which is accurately known in terms of the international kilogram can be given to the student as an "unknown." It will be noticed that such a weight will use all of the weights in a set, up to but not including the 50 g. Hence in determining the weight, all of the corrections will enter in. If the weight is reported correctly it is highly probable that the standardization has been accurate, furthermore the variation affords a measure of the quality of the work. This method of course necessitates for the instructor's use, a set of weights standardized by the Bureau of Standards. I n general, however, a set will not be standardized in terms of the international kilogram but in terms of some unit such that the corrections for the entire set shall be as small as possible. Checking may be camed out as follows without the use of Bureau of Standards' weights. Three weights are given to the student of approximately 51.01 g. (A), 49.99 g. (B), and 0.999 g. (C). The student is instructed to find the corrected weight of each of these and also t o weigh (B plus 1 g. from this set) (D). Corrected weight D minus the corrected value for 1 g. should equal corrected weight B. This is merely a check for the use of the student. The ratio (A-B)/C should be reported by the student. A comparison of this value with the value obtained by the instructor using any accurately standardized set, furnishes a measure of the accuracy of the calibration. Zero to 0.01% variation from the correct ratio shows excellent work, while a variation of more than 0.05% shows a gross error. 'Revised reprints now being furnished to the students at Harvard Chemical Laboratory do contain a column for the rider readings. ' J. Am. Chcm. Soc., 42 2528 (1920).
134
JOURNAL OP CHEMICAL EDUCATION
FEBRUA~Y, 1925
Briefly stated, the principle of Richards' method for the standardization of weights is as follows: The weights in a set may be compared among themselves in terms of any units whatsoever. The size of the unit may be altered so as to make the values conform closely to the face values of the weights, any variation in proportionality being taken care of by means of small corrections which can under certain conditions be i n terms of the original zmit i n which the arbitrarily chosen preliminary standard w m expressed. A mathematical proof is given in order to emphasize that the correction as determined may be applied without appreciable error directly to the face value of the weight. Let w, he the weight of the smallest weight to be calibrated in the set. This may be in any units whatsoever. Consider that the beam of the balance is divided into proportional parts, decimal parts for simplicity, and that the weight of the rider is also wl or that it differs from WI by a negligible amount. Now with w, on the left-hand pan, counterpoise with any material whatsoever. Another set of weights makes a convenient counterpoise. Use the rider to secure the final balance being sure that it is approximately in the center of the graduated beam in order to allow for corrections. Now withdraw w, and replace it with wz of approximately the same mass. h change of ez of the rider on the beam will again secure a balance. Hence, since the beam is a proportional scale: w, = m,
Repeating with another weight, wz: w 8 = zu,
+ e*
+e
Place w, and w2 togethe; on the left-hand pan of the balance and counterpoise as before. Withdraw the two and replace them with wa of mass approximately twice that of wl. A change of e6 on the beam will again secure a balance: ud = w, + w 2 el = 2w + er + e4 = 2w1 + E4
+
and El = e.
+
e4
This process can be repeated until we have corrections to be applied to all of the weights in the set. For any weight: w, = m,v,
+ E,,
and E,
= Ze
as determined. This means that w, ism, times as heavy as the weight w,, plus a correction, E,, all in terms of the preliminary unit of weight.
We might wish to choose wk as our final standard. uk
mkwl
=
+ Ek
I t is important to notice a t this point that the weights in terms of w, after applying the corrections as determined, are exactly proportional: w , - En = nnw and w x - EL = mkwt The ratio between the ~orrectedweights is m,/mk. Unfortunately, wk as measured in these preliminary units, does not correspond to the face value of the weight. Thus, if we wish to choose a weight marked 10 for our final standard, we might find that it had a value of 10.0184 in m, units. Hence, the w~ unit which we chose as a preliminary standard was too small and if a unit 100,184/100,000 times as large had been chosen, then the face value of the 10 would have come out all right. Let W k= face value of k weight.
or W kin units of the face value of the k weight, equals the weight in w l units times a certain factor. Since this conversion factor will always 6). have to be determined experimentally we may just as well call it (1
+
Or, the conversion factor from w1 units to Wk units is equal to the ratio of the iace value of the k weight to the observed value in w, units. Now in these new units, i. e . , in terms of the face value of the k weight: Ma
( W & = - (mn7u1
Mk
+ Ex)(L + 6 )
Theoretically
The correction to be applied is - (W& - (W& =
(-M" Ek - E , ) ( 1 + 6) Mi
Since the correction in w lunits is merely between the two corrections is in wl units:
If, as is usually the case, the E corrections are nearly proportional to the size of the weight4 and if 6 be small, then the product of the two becomes exceedingly small. ~
E,
M"
E,
M,
Assuming exact proportionality - = -,
M.
and -E,-E,
Mk
=0
Refemng t o the previous problem, a weight marked 1was found t o weigh 1.0016 in w, units. If its face value were correct, it should weigh onetenth as much as the weight marked 10 or in wl units it should weigh 1.00184. Hence a correction of 0.00024 in w1 units would have to he aoolied.'
The error in the correction6 as determined above is
which is within the experimental error and entirely negligible. Summary
1. A data sheet is given for use with the Richards' method of standardization of weights in which all of the data and computations are included on a single sheet. 2. A method is described for checking the accuracy of the calibration of a set of weights. 3. A mathematical proof is given for the Richards' method of standardizing weights. Conditions are determined for which the corrections in terms of the preliminary unit, may without appreciable error, be applied to the face values of the weights.
